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Definition df-r1 7679
 Description: Define the cumulative hierarchy of sets function, using Takeuti and Zaring's notation (). Starting with the empty set, this function builds up layers of sets where the next layer is the power set of the previous layer (and the union of previous layers when the argument is a limit ordinal). Using the Axiom of Regularity, we can show that any set whatsoever belongs to one of the layers of this hierarchy (see tz9.13 7706). Our definition expresses Definition 9.9 of [TakeutiZaring] p. 76 in a closed form, from which we derive the recursive definition as theorems r10 7683, r1suc 7685, and r1lim 7687. Theorem r1val1 7701 shows a recursive definition that works for all values, and theorems r1val2 7752 and r1val3 7753 show the value expressed in terms of rank. Other notations for this function are R with the argument as a subscript (Equation 3.1 of [BellMachover] p. 477), with a subscript (Definition of [Enderton] p. 202), M with a subscript (Definition 15.19 of [Monk1] p. 113), the capital Greek letter psi (Definition of [Mendelson] p. 281), and bold-face R (Definition 2.1 of [Kunen] p. 95). (Contributed by NM, 2-Sep-2003.)
Assertion
Ref Expression
df-r1

Detailed syntax breakdown of Definition df-r1
StepHypRef Expression
1 cr1 7677 . 2
2 vx . . . 4
3 cvv 2948 . . . 4
42cv 1651 . . . . 5
54cpw 3791 . . . 4
62, 3, 5cmpt 4258 . . 3
7 c0 3620 . . 3
86, 7crdg 6658 . 2
91, 8wceq 1652 1
 Colors of variables: wff set class This definition is referenced by:  r1funlim  7681  r1fnon  7682  r10  7683  r1sucg  7684  r1limg  7686
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